Separable algebra explained

In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.

Definition and first properties

A homomorphism of (unital, but not necessarily commutative) rings

K \to A is called separable if the multiplication map\begin{array}{rccc} \mu :& A ⊗ _{K} A &\to& A \\ & a ⊗ b &\mapsto & ab \end{array} admits a section\sigma: A \to A ⊗ _{K} A that is a homomorphism of A-A-bimodules.

If the ring

K is commutative and K \to A maps K into the center of A , we call A a separable algebra over K .

It is useful to describe separability in terms of the element

p : = \sigma( 1 ) = \sum a _{i} ⊗ b _{i} \in A ⊗ _{K} A

The reason is that a section σ is determined by this element. The condition that σ is a section of μ is equivalent to

\sum a _{i} b _{i} = 1 and the condition that σ is a homomorphism of A-A-bimodules is equivalent to the following requirement for any a in A:\sum a a _{i} ⊗ b _{i} = \sum a _{i} ⊗ b _{i} a. Such an element p is called a separability idempotent, since regarded as an element of the algebra A ⊗ A ^{\rm} it satisfies p ^{2} = p .

Examples

M _{n(R)} is a separable R-algebra. For any 1 \le j \le n , a separability idempotent is given by $\backslash sum\_^n\; e\_\; \backslash otimes\; e\_$, where e _{ij} denotes the elementary matrix which is 0 except for the entry in the entry, which is 1. In particular, this shows that separability idempotents need not be unique.

Separable algebras over a field

a with irreducible polynomial $p(x)\; =\; (x\; -\; a)\; \backslash sum\_^\; b\_i\; x^i$, then a separability idempotent is given by $\backslash sum\_^\; a^i\; \backslash otimes\_K\; \backslash frac$. The tensorands are dual bases for the trace map: if $\backslash sigma\_1,\backslash ldots,\backslash sigma\_$ are the distinct K-monomorphisms of L into an algebraic closure of K, the trace mapping Tr of L into K is defined by $Tr(x)\; =\; \backslash sum\_^\; \backslash sigma\_i(x)$. The trace map and its dual bases make explicit L as a Frobenius algebra over K.

More generally, separable algebras over a field K can be classified as follows: they are the same as finite products of matrix algebras over finite-dimensional division algebras whose centers are finite-dimensional separable field extensions of the field K. In particular: Every separable algebra is itself finite-dimensional. If K is a perfect field – for example a field of characteristic zero, or a finite field, or an algebraically closed field – then every extension of K is separable, so that separable K-algebras are finite products of matrix algebras over finite-dimensional division algebras over field K. In other words, if K is a perfect field, there is no difference between a separable algebra over K and a finite-dimensional semisimple algebra over K.It can be shown by a generalized theorem of Maschke that an associative K-algebra A is separable if for every field extension $L/K$ the algebra $A\backslash otimes\_K\; L$ is semisimple.

Group rings

If K is commutative ring and G is a finite group such that the order of G is invertible in K, then the group algebra K[''G''] is a separable K-algebra. A separability idempotent is given by $\backslash frac\; \backslash sum\_\; g\; \backslash otimes\; g^$.

Equivalent characterizations of separability

There are several equivalent definitions of separable algebras. A K-algebra A is separable if and only if it is projective when considered as a left module of

A ^{e} in the usual way. Moreover, an algebra A is separable if and only if it is flat when considered as a right module of A ^{e} in the usual way.

Separable algebras can also be characterized by means of split extensions: A is separable over K if and only if all short exact sequences of A-A-bimodules that are split as A-K-bimodules also split as A-A-bimodules. Indeed, this condition is necessary since the multiplication mapping $\backslash mu\; :\; A\; \backslash otimes\_K\; A\; \backslash rightarrow\; A$ arising in the definition above is a A-A-bimodule epimorphism, which is split as an A-K-bimodule map by the right inverse mapping $A\; \backslash rightarrow\; A\; \backslash otimes\_K\; A$ given by

a \mapsto a ⊗ 1 . The converse can be proven by a judicious use of the separability idempotent (similarly to the proof of Maschke's theorem, applying its components within and without the splitting maps).

Equivalently, the relative Hochschild cohomology groups

H ^{n(R,S;M)} of in any coefficient bimodule M is zero for . Examples of separable extensions are many including first separable algebras where R is a separable algebra and S = 1 times the ground field. Any ring R with elements a and b satisfying, but ba different from 1, is a separable extension over the subring S generated by 1 and bRa.

Relation to Frobenius algebras

A separable algebra is said to be strongly separable if there exists a separability idempotent that is symmetric, meaning

e = \sum ^{n} _{i =1} x _{i} ⊗ y _{i} = \sum ^{n} _{i =1} y _{i} ⊗ x _{i}

An algebra is strongly separable if and only if its trace form is nondegenerate, thus making the algebra into a particular kind of Frobenius algebra called a symmetric algebra (not to be confused with the symmetric algebra arising as the quotient of the tensor algebra).

If K is commutative, A is a finitely generated projective separable K-module, then A is a symmetric Frobenius algebra.

Relation to formally unramified and formally étale extensions

Any separable extension of commutative rings is formally unramified. The converse holds if A is a finitely generated K-algebra. A separable flat (commutative) K-algebra A is formally étale.

Further results

A theorem in the area is that of J. Cuadra that a separable Hopf–Galois extension has finitely generated natural R. A fundamental fact about a separable extension is that it is left or right semisimple extension: a short exact sequence of left or right that is split as, is split as . In terms of G. Hochschild's relative homological algebra, one says that all are relative -projective. Usually relative properties of subrings or ring extensions, such as the notion of separable extension, serve to promote theorems that say that the over-ring shares a property of the subring. For example, a separable extension R of a semisimple algebra S has R semisimple, which follows from the preceding discussion.

There is the celebrated Jans theorem that a finite group algebra A over a field of characteristic p is of finite representation type if and only if its Sylow p-subgroup is cyclic: the clearest proof is to note this fact for p-groups, then note that the group algebra is a separable extension of its Sylow p-subgroup algebra B as the index is coprime to the characteristic. The separability condition above will imply every finitely generated M is isomorphic to a direct summand in its restricted, induced module. But if B has finite representation type, the restricted module is uniquely a direct sum of multiples of finitely many indecomposables, which induce to a finite number of constituent indecomposable modules of which M is a direct sum. Hence A is of finite representation type if B is. The converse is proven by a similar argument noting that every subgroup algebra B is a direct summand of a group algebra A.

References

• Book: DeMeyer . F. . Ingraham . E. . Separable algebras over commutative rings . Lecture Notes in Mathematics . 181 . . Berlin-Heidelberg-New York . 1971 . 978-3-540-05371-2 . 0215.36602 .
• Samuel Eilenberg and Tadasi Nakayama, On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings, Nagoya Math. J. Volume 9 (1955), 1–16.